Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours

Hongxia Sun; Yao Wan; Yu Li; Linlin Zhang; Zhen Zhou

doi:10.3934/jimo.2019125

Article Contents

2021, Volume 17, Issue 2: 601-631. Doi: 10.3934/jimo.2019125

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Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours

1.
Business School, Beijing Technology and Business University, Beijing 100048, China
2.
State Grid Beijing Logistic Supply Company, State Grid Beijing Electric Power Company, Beijing 100054, China
3.
School of Management, Capital Normal University, Beijing 100048, China

^* Corresponding author: Hongxia Sun
^* Corresponding author: Hongxia Sun

Received: September 30, 2018

Revised: April 30, 2019

Early access: October 2019

Published: March 2021

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Abstract

We study competition in a dual-channel supply chain in which a single supplier sells a single product through its own direct channel and through two different duopolistic retailers. The two retailers have three competitive behaviour patterns: Cournot, Collusion and Stackelberg. Three models are respectively constructed for these patterns, and the optimal decisions for the three patterns are obtained. These optimal solutions are compared, and the effects of certain parameters on the optimal solutions are examined for the three patterns by considering two scenarios: a special case and a general case. In the special case, the equilibrium supply chain structures are analysed, and the optimal quantity and profit are compared for the three different competitive behaviours. Furthermore, both parametric and numerical analyses are presented, and some managerial insights are obtained. We find that in the special case, the Stackelberg game allows the supplier to earn the highest profit, the retailer playing the Collusion game makes the supplier earn the lowest profit, and the Stackelberg leader can gain a first-mover advantage as to the follower. In the general case, the supplier can achieve a higher profit by raising the maximum retail price or holding down the self-price sensitivity factor.

Keywords:

Mathematics Subject Classification: Primary: 90B50; Secondary: 91A10, 91A40.

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Figure 1. Supply chain structure

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Figure 2. Effect of $a_k$ on $\Pi_0$ and ($\Pi_1$-$\Pi_2$) in three patterns

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Figure 3. Effect of $\theta_k$ on $q_0$ and $ \Pi_0$ in three patterns

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Figure 4. Effect of $\theta_k$ on $q_1$ and $\Pi_1$ in three patterns

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Figure 5. Effect of $\theta_k$ on $q_2$ and $\Pi_2$ in three patterns

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Figure 6. Effect of $\theta_k$ or $\beta$ on ($\Pi_1$-$\Pi_2$) in three patterns

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Figure 7. Effect of $\beta$ on $q_k$ or $\Pi_k$ in three patterns

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Table 1. The equilibrium supply chain structure about the three different competitive behaviors

Structure	Wholesale supplier	Dual-channel	Monopoly retailer
Cournot	$ \delta\leq\frac{2\beta}{2\theta+\beta} $	$ \frac{2\beta}{2\theta+\beta}<\delta<\frac{\theta_0}{\beta} $	$ \delta\geq\frac{\theta_0}{\beta} $
$ q^{co}_0 $	-	$ \frac{2\beta(a-c)-a_0(2\theta+\beta)}{2[2\beta^2-\theta_0(2\theta+\beta)]} $	$ \frac{a_0}{2\theta_0} $
$ q^{co}_1 $	$ \frac{a-c}{2(2\theta+\beta)} $	$ \frac{V}{2[2\beta^2-\theta_0(2\theta+\beta)]} $	-
$ q^{co}_2 $	$ \frac{a-c}{2(2\theta+\beta)} $	$ \frac{V}{2[2\beta^2-\theta_0(2\theta+\beta)]} $	-
$ w^{co} $	$ \frac{a+c}{2} $	$ \frac{a+c}{2} $	-
$ p_0^{co} $	-	$ \frac{a_0}{2} $	$ \frac{a_0}{2} $
$ p_1^{co} $	$ \frac{(a+c)(\theta+\beta)+2a\theta}{2(2\theta+\beta)} $	$ \frac{a+c}{2}+\frac{V\theta}{2[2\beta^2-\theta_0(2\theta+\beta)]} $	-
$ p_2^{co} $	$ \frac{(a+c)(\theta+\beta)+2a\theta}{2(2\theta+\beta)} $	$ \frac{a+c}{2}+\frac{V\theta}{2[2\beta^2-\theta_0(2\theta+\beta)]} $	-
$ \Pi_0^{co} $	$ \frac{(a-c)^2}{2(2\theta+\beta)} $	$ \frac{2(a-c)(a_0\beta+V)-a_0^2(2\theta+\beta)}{4[2\beta^2-\theta_0(2\theta+\beta)]} $	$ \frac{a_0^2}{4\theta_0} $
$ \Pi_{1}^{co} $	$ \frac{\theta(a-c)^2}{4(2\theta+\beta)^2} $	$ \frac{\theta V^2}{4[2\beta^2-\theta_0(2\theta+\beta)]^2} $	-
$ \Pi_{2}^{co} $	$ \frac{\theta(a-c)^2}{4(2\theta+\beta)^2} $	$ \frac{\theta V^2}{4[2\beta^2-\theta_0(2\theta+\beta)]^2} $	-
Collusion	$ \delta\leq\frac{\beta}{\theta+\beta} $	$ \frac{\beta}{\theta+\beta}<\delta<\frac{\theta_0}{\beta} $	$ \delta\geq\frac{\theta_0}{\beta} $
$ q^{cn}_0 $	-	$ \frac{a_0(\theta+\beta)-\beta(a-c)}{2[\theta_0(\theta+\beta)-\beta^2]} $	$ \frac{a_0}{2\theta_0} $
$ q^{cn}_1 $	$ \frac{a-c}{4(\theta+\beta)} $	$ \frac{V}{4[\beta^2-\theta_0(\theta+\beta)]} $	-
$ q^{cn}_2 $	$ \frac{a-c}{4(\theta+\beta)} $	$ \frac{V}{4[\beta^2-\theta_0(\theta+\beta)]} $	-
$ w^{cn} $	$ \frac{a+c}{2} $	$ \frac{a+c}{2} $	-
$ p_0^{cn} $	-	$ \frac{a_0}{2} $	$ \frac{a_0}{2} $
$ p_1^{cn} $	$ \frac{3a+c}{4} $	$ \frac{a+c}{2}+\frac{(\theta+\beta)V}{4[\beta^2-\theta_0(\theta+\beta)]} $	-
$ p_2^{cn} $	$ \frac{3a+c}{4} $	$ \frac{a+c}{2}+\frac{(\theta+\beta)V}{4[\beta^2-\theta_0(\theta+\beta)]} $	-
$ \Pi_0^{cn} $	$ \frac{(a-c)^2}{4(\theta+\beta)} $	$ \frac{(a-c)(V+a_0\beta)-a_0^2(\theta+\beta)}{4[\beta^2-\theta_0(\theta+\beta)]} $	$ \frac{a_0^2}{4\theta_0} $
$ \Pi_{1}^{cn} $	$ \frac{(a-c)^2}{16(\theta+\beta)} $	$ \frac{V^2(\theta+\beta)}{16[\beta^2-\theta_0(\theta+\beta)]^2} $	-
$ \Pi_{2}^{cn} $	$ \frac{(a-c)^2}{16(\theta+\beta)} $	$ \frac{V^2(\theta+\beta)}{16[\beta^2-\theta_0(\theta+\beta)]^2} $	-
Stackelberg	$ \delta\leq\frac{\beta U}{T} $	$ \frac{\beta U}{T}<\delta<\frac{\theta_0}{\beta} $	$ \delta\geq\frac{\theta_0}{\beta} $
$ q^{st}_0 $	-	$ \frac{\beta U(a-c)-a_0T}{2(\beta^2 U-\theta_0T)} $	$ \frac{a_0}{2\theta_0} $
$ q^{st}_1 $	$ \frac{\theta(a-c)(2\theta-\beta)}{T} $	$ \frac{\theta V(2\theta-\beta)}{\beta^2 U-\theta_0T} $	-
$ q^{st}_2 $	$ \frac{(a-c)(4\theta^2-\beta^2-2\theta\beta)}{2T} $	$ \frac{V(4\theta^2-\beta^2-2\theta\beta)}{2(\beta^2 U-\theta_0T)} $	-
$ w^{st} $	$ \frac{a+c}{2} $	$ \frac{a+c}{2} $	-
$ p_0^{st} $	-	$ \frac{a_0}{2} $	$ \frac{a_0}{2} $
$ p_1^{st} $	$ a-\frac{(a-c)(4\theta^3+2\theta^2\beta-\beta^3-2\theta\beta^2)}{2T} $	$ \frac{a+c}{2}+\frac{V(2\theta^2-\beta^2)(2\theta-\beta)}{2(\beta^2 U-\theta_0T)} $	-
$ p_2^{st} $	$ a-\frac{\theta(a-c)(4\theta^2+2\theta\beta-3\beta^2)}{2T} $	$ \frac{a+c}{2}+\frac{\theta V(4\theta^2-\beta^2-2\theta\beta)}{2(\beta^2 U-\theta_0T)} $	-
$ \Pi_0^{st} $	$ \frac{U(a-c)^2}{4T} $	$ \frac{U(a-c)(a_0\beta+V)-a_0^2T}{4(\beta^2 U-\theta_0T)} $	$ \frac{a_0^2}{4\theta_0} $
$ \Pi_{1}^{st} $	$ \frac{\theta(a-c)^2(2\theta-\beta)(4\theta^3-2\theta^2\beta+\beta^3-2\theta\beta^2)}{2T^2} $	$ \frac{T(2\theta-\beta)^2V^2}{8(\beta^2 U-\theta_0T)^2} $	-
$ \Pi_{2}^{st} $	$ \frac{\theta(a-c)^2(4\theta^2-\beta^2-2\theta\beta)^2}{4T^2} $	$ \frac{\theta V^2(4\theta^2-2\theta\beta-\beta^2)^2}{4(\beta^2 U-\theta_0T)^2} $	-
where $ T = 4\theta(2\theta^2-\beta^2) $, $ U = 8\theta^2-4\theta\beta-\beta^2 $, $ V = a_0\beta-\theta_0(a-c) $

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Table 2. Partial derivatives of optimal quantity with respect to $a_k$ in three patterns

Cournot	$a_0$	$a_1$	$a_2$
$q^{co}_0$	$\frac{-E}{2P}$	$\frac{\beta(2\theta_2-\beta)}{2P}$	$\frac{\beta(2\theta_1-\beta)}{2P}$
$q^{co}_1$	$\frac{-\beta(\beta-2\theta_2)}{2P}$	$\frac{P(2\theta_2D+E)-\beta^2(2\theta_2-\beta)^2D}{2PDE}$	$\frac{-P(\beta D+E)-\beta^2(2\theta_1-\beta)(2\theta_2-\beta)D}{2PDE}$
$q^{co}_2$	$\frac{-\beta(\beta-2\theta_1)}{2P}$	$\frac{-P(\beta D+E)-\beta^2(2\theta_1-\beta)(2\theta_2-\beta)D}{2PDE}$	$\frac{P(2\theta_1D+E)-\beta^2(2\theta_1-\beta)^2D}{2PDE}$
Collusion	$a_0$	$a_1$	$a_2$
$q^{cn}_0$	$\frac{\theta_1\theta_2-\beta^2}{Q}$	$\frac{-\beta(\theta_2-\beta)}{2Q}$	$\frac{-\beta(\theta_1-\beta)}{2Q}$
$q^{cn}_1$	$\frac{\beta(\beta-\theta_2)}{2Q}$	$\frac{Q(2\theta_2G+H)+2G\beta^2(\theta_2-\beta)^2}{4QGH}$	$\frac{-Q(2\beta G+H)+2G\beta^2(\theta_1-\beta)(\theta_2-\beta)}{4QGH}$
$q^{cn}_2$	$\frac{\beta(\beta-\theta_1)}{2Q}$	$\frac{-Q(2\beta G+H)+2G\beta^2(\theta_1-\beta)(\theta_2-\beta)}{4QGH}$	$\frac{Q(2\theta_1G+H)+2G\beta^2(\theta_2-\beta)^2}{4QGH}$
Stackelberg	$a_0$	$a_1$	$a_2$
$q^{st}_0$	$\frac{2M\theta_2}{R}$	$\frac{-\theta_2\beta(2\theta_2-\beta)}{R}$	$\frac{-\beta S}{2R}$
$q^{st}_1$	$\frac{\theta_2\beta(\beta-2\theta_2)}{R}$	$\frac{R(\theta_2 N+2\theta_2M)+\theta_2\beta^2(2\theta_2-\beta)^2N}{2RMN}$	$\frac{-R(4M\theta_2+\beta N)+\beta^2(2\theta_2-\beta)SN}{4RMN}$
$q^{st}_2$	$\frac{\beta(2\theta_2\beta-E)}{2R}$	$\frac{-R(4M\theta_2+\beta N)+\beta^2(2\theta_2-\beta)SN}{4RMN}$	$\frac{RNE+S^2(\beta^2N-2M\theta_0\theta_2)}{4\theta_2RMN}$
where $P=\beta^2D-\theta_0E$, $Q=\theta_0H-\beta^2G$, $R=4M\theta_0\theta_2-\beta^2N$, $S=E-2\theta_2\beta$.

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Table 3. The optimal solutions for three different competitive behaviors

Optimal	$q_0$	$q_1 $	$q_2 $	$w$	$p_0$	$p_1$	$p_2$	$\Pi_0$	$\Pi_1$	$\Pi_2$
Cournot	$4.83$	$3.44$	$2.90$	$14.25$	$8.00$	$17.69$	$20.06$	$116.36$	$11.85$	$16.86$
Collusion	$5.29$	$2.81$	$2.60$	$14.13$	$8.00$	$18.24$	$20.74$	$108.01$	$11.57$	$17.23$
Stackelberg	$4.76$	$3.58$	$2.90$	$14.23$	$8.00$	$17.59$	$20.03$	$117.35$	$12.04$	$16.82$

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